MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gagrp Structured version   Visualization version   GIF version

Theorem gagrp 19141
Description: The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
gagrp ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Proof of Theorem gagrp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2733 . . . 4 (+g𝐺) = (+g𝐺)
3 eqid 2733 . . . 4 (0g𝐺) = (0g𝐺)
41, 2, 3isga 19140 . . 3 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (((0g𝐺) 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simplbi 499 . 2 ( ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
65simpld 496 1 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475   × cxp 5670  wf 6531  cfv 6535  (class class class)co 7396  Basecbs 17131  +gcplusg 17184  0gc0g 17372  Grpcgrp 18806   GrpAct cga 19138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-map 8810  df-ga 19139
This theorem is referenced by:  gafo  19145  gass  19150  galcan  19153  gacan  19154  gapm  19155  gaorber  19157  gastacl  19158  galactghm  19256  sylow2alem2  19470
  Copyright terms: Public domain W3C validator